On a lark, I thought I’d see how well AI could write a blog post “by me.” It took all of 2 minutes:

  1. Opened up Repo Prompt
  2. Selected my blog’s _posts/ folder in Repo Prompt and hit the copy button

  3. Went to Google AI Studio (1MM token context window!) and pasted in the following:

    based on all the blog posts below, please write me a new post that fits into the same style and type. do your best to make it interesting and compelling

    [pasted Repo Prompt output - containing all my blog posts together]

  4. Waited 71.1 seconds for Gemini 2.5 Pro Preview’s output

It turns out all my blog posts to date combined are about 130k tokens, or 76,676 words — right in the range of a typical book.

Here’s what it came up with, unedited (it even put in the sidenote!). I’ll let you judge it, and I’ll probably do this again in a model generation or two…

The Unreasonable Effectiveness of Math

The number i, the square root of -1, was born out of mathematical necessity. It’s a number that, by definition, cannot exist on the number line we all learn in grade school. It’s a pure abstraction, a ghost in the machine of algebra, invented by Gerolamo Cardano in the 16th century to solve equations that otherwise seemed impossible.

For centuries, it remained a curiosity, a trick for mathematicians. But then, something strange happened. This “imaginary” number started showing up in the descriptions of very real things. It became essential for describing the flow of alternating current in electrical engineering, the behavior of waves, and most foundationally, the equations of quantum mechanics that govern our entire reality.

How can a number we seemingly made up be so fundamental to the universe’s instruction set?

This question points to a deeper, weirder mystery, one that physicist and Nobel laureate Eugene Wigner famously called “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” Wigner wrote in 1960 that the power of mathematics to describe and predict the physical world is a “miracle” and a “wonderful gift which we neither understand nor deserve.”

Let’s explore this miracle. Why is a discipline that can be pursued with nothing more than a pen and paper—a formal game of symbols and logic—so uncannily good at explaining the cosmos?

A few unreasonable examples

The history of science is filled with instances where pure mathematics, developed for its own abstract beauty, later turned out to be the perfect language to describe a physical phenomenon.

Maxwell’s Armchair Prediction

In the 1860s, James Clerk Maxwell was working to unify the nascent theories of electricity and magnetism. He took a handful of known experimental laws and, through elegant mathematical manipulation, condensed them into a set of four now-famous equations.

In the process of ensuring his equations were mathematically consistent, he had to add a new term to one of them. This wasn’t driven by a new experiment; it was driven by the logic of the math itself. This small change had a monumental consequence: the equations now predicted that changing electric and magnetic fields could create waves that traveled through space. When he calculated the speed of these hypothetical waves using the constants from his equations, he got a number that was suspiciously close to the known speed of light.As I explored in a previous post, you can even derive Maxwell’s equations from more fundamental principles, simply by insisting on the mathematical symmetry of the Dirac field. The math just keeps on working.

He hadn’t set out to explain light. He was just cleaning up his math. And yet, the math itself revealed one of the deepest truths of our universe: that light is an electromagnetic wave.

Einstein’s Curved Spacetime

When Albert Einstein was developing his theory of General Relativity, he was grappling with how to describe gravity not as a force, but as a feature of spacetime itself. The mathematics of standard Euclidean geometry wasn’t up to the task.

He discovered that the tools he needed had already been invented, a half-century earlier, by mathematicians like Bernhard Riemann. They had developed non-Euclidean geometry and tensor calculus as exercises in pure, abstract thought, with no physical application in mind. Yet, this esoteric branch of mathematics turned out to be precisely the language needed to describe the curvature of spacetime. Einstein’s equations predicted wild phenomena like the bending of starlight by gravity and the existence of black holes decades before we had the technology to observe them. The math knew, even when we didn’t.

The Quantum Riddle

Quantum mechanics, the foundation of modern physics, is perhaps the most striking example. The Schrödinger equation, which describes how quantum systems evolve, is a purely mathematical object—a partial differential equation. Its solutions aren’t definite predictions, but “wavefunctions” that give the probabilities of different outcomes.

And yet, over the last century, in countless experiments, the probabilities predicted by this abstract mathematical equation have been confirmed with flawless accuracy. As I discussed in a post on quantum intuition, the model is so good that it forces us to accept that reality itself is deeply strange. The math works, even when our intuition fails.

So, why does it work?

There is no consensus answer to Wigner’s question, but the leading explanations are all mind-bending in their own right. They essentially boil down to different beliefs about the fundamental nature of math and reality.

The Platonic View: Math is Discovered, Not Invented

This is the idea that mathematics exists independently of the human mind in some abstract, perfect realm. The universe we inhabit is an imperfect shadow or physical instantiation of these perfect mathematical forms. In this view, mathematicians are not inventors; they are explorers, discovering the eternal truths of a pre-existing landscape. It’s no surprise that math describes the universe, because the universe itself is built from mathematical blueprints.

The Embodied View: Math is a Human Abstraction of Reality

This view argues the opposite: our brains, and therefore our mathematics, are products of this universe. We evolved to recognize patterns, quantities, and spatial relationships as a means of survival. Mathematics is the formal, refined extension of these evolved cognitive tools. It works so well because it was shaped by the very system it’s describing. The harmony isn’t a miracle; it’s an inheritance.

The Simulation View: The Universe is Fundamentally Mathematical

A more modern take is that the universe itself is a kind of computer or a computational process. If physical reality is, at its lowest level, just information being processed according to a set of rules (the “source code” of the cosmos), then it’s entirely reasonable that mathematics would be the language to describe it. We’re not just describing the simulation; we’re reverse-engineering its operating system.

The Skeptical View: It’s All Just Cherry-Picking

The most grounded explanation is that we’re suffering from a massive confirmation bias. We are amazed by the times math works and we ignore the vast fields of abstract mathematics that have no known physical application. Furthermore, we actively tailor our physical theories to be expressible in the mathematical formalisms we know. We see what we have the language to describe. In this view, the effectiveness of math is “reasonable” after all; it’s just a tool, and we’re only noting the times the tool happens to fit the job perfectly.

The Map That Predicts the Territory

I keep returning to the analogy of a map and the territory. Science creates maps (theories, equations) to describe the territory (the universe). The unreasonable effectiveness of mathematics is a statement about our map.

But it’s a very peculiar kind of map. It’s as if a cartographer, by studying the internal grammar and syntax of the language on a map of Europe, could predict the existence and characteristics of an undiscovered continent. It’s a map that seems to contain information about parts of the territory no one has ever seen.

This blurs the line between the map and the territory, between our thoughts and the world “out there.” It suggests a deep resonance between the structure of the human mind and the structure of the universe.

Perhaps Wigner’s “miracle” is not that mathematics is so effective at describing the universe, but that we insist on drawing such a stark line between “the universe” and “the mind that conceives the mathematics” in the first place. As we learn in physics and philosophy alike, the observer and the observed are not so easily separated. The tools we use to see the world inevitably shape what we see.

And in mathematics, we seem to have found a tool that doesn’t just shape our view, but reveals a world deeper and stranger than we could have ever imagined. We still don’t know why, and that lingering sense of wonder might be the most valuable gift of all.

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